1. Field of the Invention
The present invention relates to a method allowing fast simulation of flows in a heterogeneous porous medium such as a petroleum reservoir or an aquifer which is efficient for all types of permeability contrast.
2. Description of the Prior Art
One of the most important properties of the porous medium is its permeability K, defined by Darcy's law (1856). Because of the heterogeneity of the porous medium, permeability K is often considered as a random variable. In fact, the porous medium is represented by a permeability field that varies in space.
The flow of a single fluid with a steady-state flow in a porous medium is governed by Darcy's law:
  U  =                    -                  K          η                    ⁢                          ⁢              grad        ⁡                  (          P          )                      =                  -                  K          η                    ⁢              ∇        P            where U is the filtration rate, K the permeability, P the pressure of the fluid and η the viscosity of the fluid. The fluid is incompressible and the mass conservation equation is written as follows:div(U)=0
By combining the two equations above, the diffusivity equation is obtained as follows:
      div    ⁡          (                        K          η                ⁢                  ∇          P                    )        =  0
Whatever the technique used to simulate flows in a heterogeneous porous medium, this diffusivity equation has to be solved. For a given permeability field discretized by a reservoir grid, the corresponding pressure and velocity fields have to be estimated.
The following documents, mentioned in the description hereafter, illustrate the state of the art:    Darcy, H., Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris, France, 1856.    Eyre, D. J., and Milton, G. W., A Fast Numerical Scheme for Computing the Response of Composites Using Grid Refinement. The European Physical Journal Applied Physics, 6:41-47, 1999.    Michel J. C., Moulinec H., and Suquet P., A computational Method Based on Augmented Lagrangians and Fast Fourier Transforms for Composites with High Contrast, Comput. Model. Eng. Sc., 1(2), 79-88, 2000.    Mizell S. A., Gutjahr A. L., and Gelhar L. W., Stochastic Analysis of Spatial Variability in Two-Dimensional Steady Groundwater Flow Assuming Stationary and Non-Stationary Heads, Water Resources Research, 18(4), 1053-1067, 1982.    Moulinec, H., and Suquet, P., A Numerical Method for Computing the Overall Response of Nonlinear Composites with Complex Microstructure. Computer Methods in Applied Mechanics and Engineering, 157:69-94, 1998.    Van Lent, T. J., Numerical Spectral Methods Applied to Flow in Highly Heterogeneous Aquifers, PhD thesis, Stanford University, 1992.
Many algorithms have been developed in the past to simulate flows in heterogeneous porous media. These algorithms essentially involve finite differences, finite volumes, hybrid mixed finite elements, conformal finite elements or spectral mixed finite elements. These techniques are very general and allow modelling of very complex flows. They are characterized in that they apply to transmissivities rather than permeabilities. To date, there is no simple and totally reliable method for estimating these transmissivities from the permeabilities.
In parallel with these various approaches, methods allowing working directly from the permeabilities have been developed. These methods favoring the use of Fourier transforms to simulate the flows were introduced by Van Lent (1992). Van Lent develops the diffusivity equation according to the small-perturbation method and proposes solving it in the frequency domain by means of an iterative approach. However, this method is suitable only for continuous and weakly heterogeneous porous media. It is not suited for simulation of the flows in a highly heterogeneous medium because of the approximations associated with the solution of the diffusivity equation developed according to the small-perturbation method. This method cannot, for example, apply to a medium containing two facies such as sandstones and clay, whose petrophysical properties are very different.
These techniques, which directly work from the permeabilities, that is without going through the transmissivities, are faster than the aforementioned complex methods, but they are limited to the description of a single-phase steady-state flow in a weakly heterogeneous porous medium. In fact, their use has remained very marginal.